Mass of rays on complete open surfaces
type: Item
instance of: scholarly article
title: Mass of rays on complete open surfaces
zbMATH Open document ID: 0731.53044
author: Katsuhiro Shiohama
author: Takashi Shioya
author: Minoru Tanaka
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: A connected, finitely connected, oriented, complete and noncompact Riemannian 2-manifold \(M\) without boundary is considered in the paper. Let \(M\) have \(k\) endpoints and let \(K\subset M\) be a compact set with the property that \(M\setminus \text{Int}\, K\) consists of \(K\) tubes \(\mathcal U_1,\ldots,\mathcal U_k\) such that each \(\mathcal U_ i\) is homeomorphic to \(\mathcal S^1\times [0,\infty)\) and that each \(\partial \mathcal U_i\) is a piecewise smooth simply closed curve. Let \(k(\partial \mathcal U_i)\) denote the curvature integral over the boundary curve \(\partial \mathcal U_i\) and \(c\) the total curvature, then the values \(s_i(M)=k(\partial \mathcal U_i)-c(\mathcal U_ i)\) are independent of the choice of the tubes. The relationship between these values and the ``mass of rays'' (i.e. measure \(\mathfrak M(A(p))\) where \(A(p)\) is the set of all unit vectors tangent to rays emanating from \(p\), \(\mathfrak M\) is the natural measure on \(\mathcal S^p(1)\subset T_pM\) induced from the Riemannian metric) is studied in the work. One of the main results is Theorem B. Assume that \(M\) admits total curvature \(c(M)\) and has \(K\) endpoints. Let \(\varphi\) be a simply closed smooth curve in \(M\) and let \[ B(t)=\{x\in M; d(x,\varphi)\leq t\},\quad S(t)=\{x\in M; d(x,\varphi)=t\}. \] If \(s_i(M)\leq 2\pi\) holds for each \(i=1,\ldots,k\) then \[ \lim_{t\to \infty}\Bigl(\int_{B(t)}\mathfrak M\circ A\, dM\Bigr)/\Bigl(\int_{B(t)}\,dM\Bigr)=\sum_{i=1}^K s^2_i(M)/(2\pi \chi(M)- c(M)), \] if \(2\pi \chi (M)-c(N)>0\); and \(\lim=0\) if \(2\pi \chi (M)-c(M)=0\), where \(\chi(M)\) is the Euler characteristic.
reviewed by: M. A. Malakhaltsev
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4209012
zbMATH Keywords: noncompact surface
zbMATH Keywords: rays
zbMATH Keywords: curvature integrals
zbMATH Keywords: mass of rays
MaRDI profile type: Publication
full work available at URL: pjm.1990.143.349
OpenAlex ID: W1982287921
DOI: 10.2140/PJM.1990.143.349
description: scientific article; zbMATH DE number 4209012